A property of span of S

As we know, {S} is a nonempty subset of a vector space {V}, then the set {W} consisting of all linear combinations of elements of {S} is a subspace of {V}.

The subspace {W} described in above fact is called the span of {S} {or the subspace generated by the elements of {S}).

We have the following theorem in the book ‘Linear algebra’ of Friedberg et al.

Theorem 1 Let {S} be a linearly independent subset of a vector space {V}, and let {v} be an element of {V} that is not in {S}. Then {S \cup \{v\}} is linearly dependent if and only if { v \in span(S)}.

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